I can't install os after erasing my HD. It says (in french) 'Le serveur de recuperation n'a pas pu etre contacte'

Try booting the laptop into Internet Recovery Mode using Command + Option + R to access the macOS 13.x Ventura installer. Unfortunately no matter which keys you use to boot for Internet Recovery Mode, some Macs may only boot into the older macOS online installer for the OS which originally shipped on the Mac from the factory. Unfortunately people have trouble access the older online macOS installers because their security certificates have expired which require setting the system date & time to 2017. If you can only boot into the macOS 10.12.x Sierra installer, then launch the Terminal app from the Utilities menu on the menu bar. Enter this command to set the system date & time to Jan. 2, 2017 @ 11:33am:

date  -u  0102113317

If there are no reported errors, then quit the Terminal app and launch Disk Utility so you can erase the whole physical SSD as GUID partition and MacOS Extended (Journaled). Quit Disk Utility and select the "Reinstall macOS" option.

You can also use your old 2013 Mac to create a bootable macOS 11.x Big Sur USB installer using the instructions in this Apple article:

Create a bootable installer for macOS - Apple Support

Performance is more a function of not running any system optimizers. Advanced Mac Cleaner, and MacKeeper are adware. You don't want to add any optimizers. You want to make sure the hard disk does not get more than 85% full. Use only software that is rated for the operating system installed. Anti-virus and firewall software are basically no needed as long as you do not fall pray to phishing attacks.

https://appuals.com/this-copy-of-the-install-os-x-el-capitan-application-cant-be-verified/

If none of these help, there is a French Apple Support Community that would have people who know what this error means exactly.

Communauté d’assistance Apple officielle